### Abstract

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, L^{♯}_{p}. (f, T) and L^{♭}_{p}. (f, T), for a weight-two modular form ∑ a_{n}q^{n} and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_{p} = 0. These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič-Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

Original language | English (US) |
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Pages (from-to) | 885-928 |

Number of pages | 44 |

Journal | Algebra and Number Theory |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Birch and Swinnerton-Dyer
- Elliptic curve
- Iwasawa Theory
- Modular form
- P-adic L-function
- Šafarevič-Tate group

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## Cite this

Sprung, F. (2017). On pairs of P-Adic L-Functions for weight-two modular forms.

*Algebra and Number Theory*,*11*(4), 885-928. https://doi.org/10.2140/ant.2017.11.885